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Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty general and apply to Laplacian-Beltrami operator on manifolds. Their paper has bounds that are possibly too loose for my application.

Thanks, John

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I have had an analogous question but received no answer: mathoverflow.net/questions/55235/… –  András Bátkai Jan 12 '13 at 9:02
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I think that Sogge's bounds are as good as as you can hope for. In any case, the answer depends on the geometry of the boundary, and more precisely the billiard dynamics. (Think off your domain as a billiad table.) –  Liviu Nicolaescu Jan 12 '13 at 12:34
    
Can you point me to the papers of Sogge that seem relevant? I see some papers involving $L_p$ bounds, but not anything on $L_\infty$. –  Ray Yang Feb 2 '13 at 6:49
    
The recent work of Fanghua Lin and Kenig for div$(a\nabla u)=\lambda u$ could be of interest for you. –  Jung Wen Chen Nov 25 '13 at 9:18

2 Answers 2

Moser iteration proceeds roughly like this: If the dimension $n$ is greater than $2$ and we assume homogeneous Dirichlet , we can proceed as follows: Using the Sobolev inequality on $\mathbb{R}^n$, $$ c(n)\|u\|_{2n/(n-2)} \le \|\nabla u\|_2 $$ and integrating by parts, we get something roughly like this (you have to figure out what to do with all the absolute values) for each $p > 1$: $$ \lambda\int |u|^p \ge \int |u|^{p-1}(-\Delta |u|) = \frac{4(p-1)}{p^2}\int |\nabla |u|^{p/2}|^2 \ge \frac{4(p-1)}{p^2}c(n)\|u\|_{np/(n-2)}^p. $$ Therefore, given $p_0 > 1$, if we set $p_k = p_0(n/(n-2))^k$, we get $$ \|u\|_{p_{k+1}} \le \left(\frac{p_k^2}{4(p_k-1)}\frac{\lambda}{c(n)}\right)^{1/p_k}\|u\|_{p_k}. $$ Iterating this, we get $$ \|u\|_\infty \le C(n,p_0)\lambda^{n/(2p_0)}\|u\|_{p_0}. $$ The power of the eigenvalue in the final estimate doesn't need to be calculated explicitly. You know what it has to be by observing that the left side is invariant under rescaling space ($\mathbb{R}^n$) and therefore the right side must be, too.

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it is possible to bound the $L^\infty$ norm of an eigenfunction in terms of its $L^2$ norm, a power of the relevant eigenvalue and a dimensional constant. this is true for every domain of $R^N$ with finite measure, no matter how bad the boundary is (I am assuming homogeneous Dirichlet boundary conditions, of course). I don't know if this answer your question.

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Do you have references for this? That would be great. –  András Bátkai Apr 24 at 14:31
    
I believe this can be done in a relatively straightforward fashion using Moser iteration. –  Deane Yang Apr 24 at 15:03
    
@DeaneYang: thank you. –  András Bátkai Apr 24 at 16:53
    
Indeed, an analogous bound also holds on fractal domains. See section 4.5 in Kigami's Analysis on Fractals. –  HMPanzo Apr 24 at 19:52
    
@DeaneYang yes, exactly. an alternative proof based on a compactness argument (which does not give an explicit constant) can be found in a paper by martin flucher, JMAA (1995) (proposition 5) –  user49954 Apr 25 at 13:46

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